Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{-9t^3 + 171t^2 - 810t}{-6t^2 + 24t + 270}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {-9t(t^2 - 19t + 90)} {-6(t^2 - 4t - 45)} $ $ p = \dfrac{9t}{6} \cdot \dfrac{t^2 - 19t + 90}{t^2 - 4t - 45} $ Simplify: $ p = \dfrac{3t}{2} \cdot \dfrac{t^2 - 19t + 90}{t^2 - 4t - 45}$ Next factor the numerator and denominator. $ p = \dfrac{3t}{2} \cdot \dfrac{(t - 9)(t - 10)}{(t - 9)(t + 5)}$ Assuming $t \neq 9$ , we can cancel the $t - 9$ $ p = \dfrac{3t}{2} \cdot \dfrac{t - 10}{t + 5}$ Therefore: $ p = \dfrac{ 3t(t - 10)}{ 2(t + 5)}$, $t \neq 9$